If (X,Ƭ) is a topological space, then a quasi-uniformity U on X is compatible with Ƭ if the quasi-uniform topology, Ƭ_u = Ƭ. This paper is concerned with local properties of quasi-uniformities on a set X that are compatible with a given topology on X. Chapter II is devoted to the construction of Hausdorff completions of transitive quasi-uniform spaces that are members of the Pervin quasi-proximity class. Chapter III discusses locally complete, locally precompact, locally symmetric and locally transitive quasi-uniform spaces. Chapter IV is devoted to function spaces of quasi-uniform spaces. Chapter V and the Appendix are concerned with the topological homeomorphism groups of quasi-uniform spaces. / Ph. D.

This paper presents new concepts in the use of infinite and infinitesimal numbers in real analysis. theory is based upon the hyperreal number system developed by Abraham Robinson in the 1960's in his invention of "nonstandard analysis". paper begins with a short exposition of the construction of the hyperreal nU1l1ber system and the fundamental results of nonstandard analysis which are used throughout the paper. The new theory which is built upon this foundation organizes the set hyperreal numbers through structures which on an infinite base logarithm. Several new relations are introduced whose properties enable the simplification of calculations involving infinite and infinitesimal The paper explores two areas of application of these results to standard problems in elementary calculus. The first is to the evaluation of limits which assume indeterminate forms. The second is to the determination of convergence of infinite series. Both applications provide methods which greatly reduce the amount of con1putation necessary in many situations. / Master of Science

In this paper we investigate several aspects of computing control system radii for finite element approximations of control systems governed by partial differential equations. Finite element approximations of the heat equation (parabolic), the wave equation (hyperbolic) and the equations of thermoelasticity (mixed) are used as test cases. Balanced realizations, reduced order models and other transformed models are also studied. / Master of Science

This thesis presents a comparison of correlation methods in risk analysis. A theoretical solution is given to the correlation problem along with a discussion of each method. Each method is compared to a developed test case and two other cost projects. Restrictions on correlation coefficients are also given followed by the advantages and disadvantages of each method. / Master of Science

We prove that no disjoint union of any number of copies of even dimensional complex projective space can bound a smooth oriented compact manifold with boundary. We prove this by defining and computing certain algebraic invariants for smooth oriented manifolds. A non-diffeomorphic relationship is established between boundary manifolds and complex projective space by contrasting invariants computed for these spaces. / Master of Science

We study elastic and inelastic (2 cluster) - (2 cluster) scattering amplitudes for N-body quantum systems. For potentials falling off like r⁻^-1-E we prove that below the lowest 3-cluster threshold these amplitudes exist, are continuous and that asymptotic completeness holds. Moreover, if potentials fall off exponentially we prove that these amplitudes can be meromorphically continued in the energy, with square root branch points at the 2 cluster thresholds. / Ph. D.

The change in the number of phase transitions for perturbations of finite range interactions is studied. A Monte-Carlo simulation was performed for a translation invariant spin 1/2 ferromagnetic model in Z² with fundamental bonds A = {(0,0);(0,1)} B = {(0,0);(2,0)} C = {(0,0);(0,1);(1,1);(1,0)} The model exhibits one phase transition if the coupling constant J(A) is zero, but two phase transitions were found when J(A) is non zero and small enough. The generalization of this situation is provided by a construction, due to J. Slawny, which through a sequence of progressively smaller perturbations yields models with an arbitrary minimum number of phase transitions. However, such construction requires the existence of interactions with one fundamental bond such that for all values of the coupling constants the Gibbs state is unique even when the interaction is perturbed by an arbitrary finite range perturbation of small enough norm. In this work it is proven that such property is exhibited by some translation invariant systems in Z^ν with finite state space at each point. The proof applies to models with real interactions and whose fundamental bonds are all multiple of a single bond which is of prime order and which is obtained as the product—in the group ring structure of the dual space—of one dimensional bonds whose non trivial projections at each lattice site are unique. The proof is based on the Dobrushin-Pecherski criterion concerning the uniqueness of Gibbs states under perturbations. Such criterion is restated so that only transition functions on sets of simple geometry are involved. In addition, an algebraic characterization is presented for the set of Gibbs states for ferromagnetic systems for which the state space at each lattice site is a compact abelian group. This is a generalization of the theory originally introduced by Slawny for spin 1/2 ferromagnetic models and later extended by Pfister to ferromagnetic models for which the state space at each point is a finite product of tori and finite abelian groups. / Ph. D.

Given the prime power factorization of a positive integer m, a method for calculating the number of all distinct n x n - involutory matrices (mod m) is derived. This is done by first developing a method for the construction and enumeration of involutory matrices (mod P^α), without duplication, for each prime power modulus P^α. Using these results, formulas for the number of distinct involutory matrices (mod P^α) of order n are given where p is an odd prime, p=2, α= 1 and α > 1. The concept of a fixed group associated with an involutory matrix (mod P^α) is used to characterize such matrices. Involutory matrices (mod P^α) of order n are considered as linear transformations on a vector space of n-tuples to provide uncomplicated proofs for the basic results concerning involutory matrices over a finite field. / Master of Science

The fundamental definitions of radiative transfer theory are given and the two-dimensional equation of transfer is derived, density of radiation is defined, and two-dimensional two-intensity transfer model is presented. An operational interpretation of the latter model is given interms of military truck transport supply and the functional dependencies of the terms in the transfer equations are evaluated. For this interpretation the density equations are given and the study state and time dependent solutions of the density equations are discussed in polar coordinates. This work was conducted for the U. S. Army Transportation Research Command, Fort Eustis, Virginia, 1961, Task 9R38-11-009-02. / Master of Science

Early in the thirteenth century, Leonardo de Pisa, or, Fibonacci, introduced his famous rabbit problem, which may be stated simply as follows: assume that rabbits reproduce at a rate such that one pair is born each month from each pair of adults not less than two months old. If one pair is present initially, and if none die, how many pairs will be present after one year? The solution to the problem gives rise to a sequence {U_n} known as the Classical Fibonacci Sequence. {U_n} is defined by the recurrence relation U_n = U_n-1 + U_n-2, n ≥ 2, U₀ = 0, U₁ = 1 Many properties of this sequence have been derived. A generalized sequence {F_n} can be obtained by retaining the law of recurrence and redefining the first two terms as F₁ = p', F₂ = p' + q' for arbitrary real numbers p' and q'. Moreover, by defining H₁ = p+iq, H₂ = r+is, p,q,r and s real, a complex sequence is determined. Hence, all the properties of the classical sequence can be extended to the complex case. By reducing the classical sequence by a modulus m, many properties of the repeating sequence that results can be derived. The Fibonacci sequence and associated golden ratio occur in communication theory, chemistry, and in nature. / Master of Science